Work & Kinetic Energy Study Notes

Work & Kinetic Energy

Prof. R. Heindl

What is Energy?

• Energy is defined as the ability to do work.

• It is a scalar quantity that is associated with the state (or condition) of one or more objects.

Types of Energy

• Mechanical Energy:

  • Kinetic: Energy that is associated with an object’s motion.

  • Potential: Energy that is associated with an object’s position.

• Thermal: Energy of the random motion of atoms within an object.

• Chemical: Energy stored in the bonds of chemical compounds.

• Solar: Energy from the sun.

• Nuclear: Energy released during nuclear reactions.

• Electric: Energy associated with electric charges.

• Magnetic: Energy related to magnetic fields.

• Important Note: Energy can be transformed from one type to another, illustrating the principle of conservation of energy.

Kinetic Energy

• Kinetic energy is defined as the energy associated with the motion of an object and is mathematically expressed as: K = \frac{1}{2} mv^2

  • Where:

  • [K] = Kinetic energy (in Joules, J)

  • m = mass of the object (in kg)

  • v = velocity of the object (in m/s)

• Kinetic energy is measured in units of:

  • kg \, m^2 \, s^{-2} = N \, m = J

• It is a scalar quantity.

New Math - Vector Product

• There are two types of products between two vectors:

  • Dot Product (Scalar Product): The result is a scalar.

  • Cross Product (Vector Product): The result is a vector.

Dot Product of Two Vectors

• For vectors \vec{A} and \vec{B}:

  • Given:

  • \vec{A} = Ax \hat{i} + Ay \hat{j}

  • \vec{B} = Bx \hat{i} + By \hat{j}

  • The dot product is defined as:
    \vec{A} \cdot \vec{B} = AB \, cos(\phi)

  • An alternate formula is:
    \vec{A} \cdot \vec{B} = AxBx + AyBy + AzBz

• The angles of the dot product result in different output values:

  • If \vec{A} \cdot \vec{B} > 0: A and B are in the same direction.

  • If \vec{A} \cdot \vec{B} = 0: A and B are perpendicular.

  • If \vec{A} \cdot \vec{B} < 0: A and B are in opposite directions.

  • Particularly, if \phi = 180^{\circ}: then \vec{A} \cdot \vec{B} = -AB.

  • If \phi = 0^{\circ}: then \vec{A} \cdot \vec{B} = AB.

Properties Related to Unit Vectors

• If calculating the dot product with unit vectors:

  • \vec{A} \cdot \hat{i} = A_x

  • \vec{A} \cdot \hat{j} = A_y

  • \hat{i} \cdot \hat{i} = 1

  • \hat{j} \cdot \hat{j} = 1

  • \hat{i} \cdot \hat{j} = 0 (they are orthogonal).

• The dot product of a vector with a unit vector gives us the projection of the vector onto the respective axis of the unit vector.

Work

• Work is defined as the energy transferred to or from an object as a result of the action of a force.

• Mathematically, work is expressed as:
  W = | \vec{F} | | \Delta r | cos(\theta) = \vec{F} \cdot \Delta \vec{r}

• Work is also a scalar quantity (like energy).

Types of Work

• Positive Work:

  • It involves the transfer of energy to the object when the force is in the direction of the displacement.

• Negative Work:

  • It represents the transfer of energy from the object when the force is opposite to the direction of displacement.

• Any force acting perpendicular to the direction of movement does zero work:
  W = \vec{F} \cdot \Delta \vec{r}

Work Done by a Variable Force

• When a variable force is applied, the differential work is given as:
  dW = \vec{F} \cdot d\vec{r}

• Subsequently, the total work done can be calculated through integration: W = \int{rA}^{r_B} \vec{F} \cdot d\vec{r}

  • It expands to:
    W = \int{xA}^{xB} Fx \, dx + \int{yA}^{yB} Fy \, dy + \int{zA}^{zB} Fz \, dz

Work-Kinetic Energy Theorem

• The theorem states that the net work done on an object equals the change in its kinetic energy: W{net} = KB - K_A = \Delta K

  • Where:

  • K{final} = K{initial} + W_{net}

Examples

• Work done by Gravitational Force (Upwards):

  • Consider a basketball tossed upwards:

  • Initial kinetic energy is given by:
    Ki = \frac{1}{2} mv^20

• Work done by Gravitational Force (Downwards):

  • The gravitational force acting on the basketball slows it down until it stops.

  • Work done by gravitational force when slowing down:
    W = \vec{F}_g \cdot d\vec{r} = mg y \cos(180^{\circ}) = -mg y

• Work During Lifting and Lowering an Object:

  • Positive work during upward displacement by the weightlifter transfers energy to the object.

  • Negative work done by gravitational force during the same action transfers energy away from the object.

• In the case of slow lowering of the weight, the relationship is:
  Kf - Ki = Wa + Wg

  • Kf = Ki = 0 thus

  • Wa = -Wg = -(-mgy) = mgy

• If the weightlifter lowers the weight slowly:

  • Wa = -Wg = -mg y leads to:

  • Kf = 0 and Ki = 0.

• Car Being Towed Example:

  • A car is towed, starting from rest; with a tension force in the tow rope, the work done can be calculated with:
    W{tot} = \int{rA}^{rB} \vec{F} \cdot d\vec{r}

• Launching a Rocket Example:

  • A 150,000 kg rocket is launched straight up.

  • The thrust generated: 4.0 \times 10^6 N for a height of 500m:

  • Work done by thrust:
    W{thrust} = F{thrust} \cdot (\Delta r) = (4.0 \times 10^6 N)(500 m) = 2.00 \times 10^9 J

  • Work done by gravity:
    W_{grav} = -mg(\Delta r) = - (1.5 \times 10^5 kg)(9.8 m/s^2)(500 m) = -0.74 \times 10^9 J

  • To find the resultant speed:
    \frac{2W}{m} = \Delta K = mv^2{f} leads to finding: v{f} = 130 m/s

System and Environment Interaction

• Interaction of system and environment:

  • System has energy ($E_{sys}$) comprised of:

  • Kinetic

  • Thermal

  • Potential

  • Chemical

• Energy transformations can occur:

  • Heat

  • Work

Work Done on a System

• Doing work on a system modifies its energy:

  • \Delta E{sys} = W{ext}

• For systems of particles interacting only via friction, the total energy becomes: E{sys} = K + E{th}

  • Where:
    \Delta E = \Delta K + \Delta E_{th} = W

  • Here, W_{tot} represents total work done.